We assume that measurements of \(Y\)  are contaminated with I. I. D. Gaussian noise.
For now, we neglect noise in \(X\) , which could be incorporated via a total least square approach [1]}, [2]},
By linearity, it is sufficient to consider \(Y_{i,j} \sim \mathcal {N}(0, 1)\) .
Again, since exact DMD and symmetric piDMD are linear models,
they are unbiased estimators:
\(\mathbb {E}\left(A^{s}_{i,j} \right)=\mathbb {E}\left(A^{e}_{i,j} \right)= 0.\) 
